Abstract

Using the elementary method and some properties of the least solution of Pell’s equation, we prove that the equation
has no positive integer solutions () with and being odd primes.

1. Introduction

Let , be the sets of all integers and positive integers, respectively. In recent years, there are many authors who investigated the various properties of exponential diophantine equation with circulating form (see [1–4]). Recently, Zhang et al. [5] are interested in the equation
Using the -adic lower bound of the log-linear model method, they proved all solutions of (1) satisfying . Meanwhile, they proposed a conjecture as follows.

Using the method in [5], it seems to be a very difficult problem to improve the upper bound estimate for . In this paper, we use the elementary method and some properties of the least solution of Pell’s equation to solve the conjecture partly. That is, we will prove the following.

Theorem 2. Equation (1) has no positive integer solution with and being odd primes.

2. Several Lemmas

Let be a nonsquare positive integer, and let denote the class number of binary quadratic primitive forms with discriminant . Then we have the following.

Lemma 3. For the equation
there is a solution with , and there is a unique positive integer solution satisfying , where pass through all positive integer solutions of (2). We call as the least solution of (2). Every solution of (2) can be expressed as

Proof. According to Table II in Chapter 16 of [6], we know that Lemma 5 holds for , and when , by Theorems and of [6], we have
where is the least solution of (2). If , and , , then by (4), we have
a contradiction. This proves Lemma 5.

Lemma 6. Let be an odd prime with . If the equation
has the solution , then every solution can be expressed as
where are positive integers satisfying
and is a solution of (2).

Let be one of the solutions of (1). Without loss of generality we may assume that , since and are symmetrical in (1). By [5], we know that are coprime, and . When and are odd primes, must be even. Note that ; by (1) we have ; then , so by the result in [5], we get

If , by (10), is an odd prime with . We see from (1) that the equation
has the solution
Since is an odd prime with , applying Lemma 6 to (11) and (12), we have
where are positive integers satisfying
is a solution of the equation
and denotes the class number of binary quadratic primitive forms with discriminant .

Since is an odd prime, we know from (13) that or . If , by (13), so from (16) we have and . But, by Lemma 5, this is impossible; thus .

Further, , and from (19) we know , since by Lemma 3. Thus, according to Lemma 3 there is such that
where is the least solution of (17). For the integer , there exist integers and satisfying
Let
From (17), (19), and (22), we know that and are integers satisfying
And from (18), (20), (21), and (22), we have

If in (21), then, from (24), we have
However, since from (9), and by (23), we have . According to (25), we get , which is impossible. Thus, from (21), we have and

Let
Then are integers with , and
where is the integral part of . From (29), we have
Applying (27) to (24), we get
From (17) and (26), , by (30), . However, we get from (9) that is an odd prime satisfying and ; then from Lemma 4, we know it is impossible. Thus, the theorem holds for .

Similarly, if , by (10) is an odd prime with . We see from (1) that the equation
has the solution
Applying Lemmas 5 and 6 to (11) and (12), we have
where are positive integers satisfying
and is a solution of the equation
Applying Lemma 3 to (34) and (35), we have
where is the least solution of (36). In addition, the integer can be expressed as
Let
From (35) and (36), we know that and are integers satisfying
And from (34), (37), (38), and (39), we get

If in (38), then, from (41), we have
Since is an odd prime, , by (42), we know
From (40), we know ; then from (43) we get . Let , and , so
By (45),
Combining (42) and (46) we may immediately get
However, from (42) and (47), we get , but it is impossible. Therefore, we have and

Now, using the similarly proof with , from (41) and (48) can obtain contradiction.

This completes the proof of our theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the P.S.F. (2013JZ001) and N.S.F. (11371291) of China. The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.